It has been desired in a hard disk to lower a flying height of a magnetic head in order to advance in the reduction in the unit recording area and increase in storage capacity. In order to lower the flying height of the head, there have strongly urged the reduction in surface roughness in the polishing step of a substrate for a hard disk, as well as the reduction in microwaviness (microwaviness being referred to as projections and dents on the surface having a longer wavelength than the surface roughness, which is referred herein to the waviness having a wavelength of from 0.5 to 5 mm), a wide range of waviness including this microwaviness [short wavelength waviness (waviness having a wavelength of from 50 to 500 μm), and long wavelength waviness (waviness having a wavelength of 0.5 mm or more)], and furthermore the reduction in expansion due to nodules (hereinafter simply referred to as nodules). In order to manufacture a substrate with reduced microwaviness and further reduced waviness and nodules, studies have been made on a multi-stage polishing in which polishing is carried out step-by-step with gradually reducing a particle size of an abrasive, and on mechanical conditions such as controlling a pore diameter of a polishing pad, hardening the hardness of the polishing pad, and controlling a polishing load and the number of rotations. However, in the formation of finer particles for the abrasive, the processing time can take a long period of time, so that the waviness cannot be reduced to the desired microwaviness in the mechanical conditions. In addition, studies have been made on a process of polishing with a polishing composition containing a water-soluble iron compound (for instance, see Japanese Patent Laid-Open No. Hei 10-204416). However, in this process, the microwaviness, and further waviness and nodules cannot said to be satisfactorily reduced.